The Best

I am not an expert in physics. Fortunately, the general concepts of pole vault physics are rather simple. Here, I present a VERY simple model with which to have fun. As an example of how any average vaulter can figure out their maximum potential height, we can do some simple math to get a ball park figure. The basic idea is this - a pole vaulter takes his/her horizontal acceleration and converts it into vertical acceleration (counteracting the acceleration of gravity) through the use of a compressible spring (that would be the pole). To derive an equation, one must first make a few assumptions to start. Okay, so you may recall that FORCE equals MASS times ACCELERATION (good ole' Newton's f=ma equation). In the case of the vault, this means that the amount of force applied to the pole at the plant equals the mass of the vaulter (his/her weight) times the vaulter's acceleration at the plant (where acceleration is the measure of change in the velocity or speed). So the first assumption we will make is that the "energy" at the bottom of the plant is converted to vertical lift without any loss. This of course does not happen in real life. No vaulter can convert 100% of the energy created at the plant into vertical lift. Much of it is lost in pole vibration, dissipation in the fiberglass, technique, etc. However, by making this assumption, we can make a mathematical model to figure out what the absolute highest height would be for a vaulter with a given weight and speed. So, we will start our model by saying that 100% of the KINETIC ENERGY gets converted to POTENTIAL ENERGY. From your intro to physics book, you know that the equation for calculating potential energy as one increases their height off the ground is:
PE = mgx